Problem
Prediction
Outcome: Juvenile Survival
Possible predictors:
- Birth date
- Birth mass
- Year
- Sex
- Maternal fecundity
- Maternal reproductive status
- Maternal age
- Population size
- Temperature (min, max, ave…)
- Rainfall
- Wind speed
- …
Hierarchy of GLMs
What is multiple regression really doing?
Determine the association of each predictor while “controlling” for the other predictors.
How?
- Allow the other variables to account for variation in the predictor of interest
- Multiple regression of the predictor of interest on the remaining predictors (response variable not involved)
- Regress the response on the residual variance of the predictor of interest
What is multiple regression really doing?
\[Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_4 X_4\]
To estimate the coefficient of \(X_1\):
- Regress \(X_1\) on \(X_2 + X_3 + X_4\)
\[X_1 = \gamma_0 + \gamma_1 X_2 + \gamma_2 X_3 + \gamma_3 X_4\]
- Calculate residuals for the model.
- Regress \(Y\) on the residuals.
- The estimated coefficient is \(\beta_1\).
Residuals
Residuals
Assumption: Residuals from OLS regression should be centered on zero and normally distributed.
Milk content across mammalian species
What are the contributions of the fat and lactose content of mammalian milk to total milk energy?
- Outcome: Kilocalories of energy per gram of milk
- Predictors:
- Percent fat
- Percent lactose
Milk content across mammalian species
milk <- read_excel("../data/Milk.xlsx", na = "NA")
glimpse(milk)
## Observations: 29 ## Variables: 8 ## $ clade <chr> "Strepsirrhine", "Strepsirrhine", "Strepsirrhine"… ## $ species <chr> "Eulemur fulvus", "E macaco", "E mongoz", "E rubr… ## $ kcal.per.g <dbl> 0.49, 0.51, 0.46, 0.48, 0.60, 0.47, 0.56, 0.89, 0… ## $ perc.fat <dbl> 16.60, 19.27, 14.11, 14.91, 27.28, 21.22, 29.66, … ## $ perc.protein <dbl> 15.42, 16.91, 16.85, 13.18, 19.50, 23.58, 23.46, … ## $ perc.lactose <dbl> 67.98, 63.82, 69.04, 71.91, 53.22, 55.20, 46.88, … ## $ mass <dbl> 1.95, 2.09, 2.51, 1.62, 2.19, 5.25, 5.37, 2.51, 0… ## $ neocortex.perc <dbl> 55.16, NA, NA, NA, NA, 64.54, 64.54, 67.64, NA, 6…
Ignore for now that these are comparative species-level data.
Wrangling data
Keep:
species: Specieskcal.per.g: Kilocalories of energy per gram of milkperc.fat: Percent fatperc.lactose: Percent lactose
Filter complete cases (drop rows with NA).
M <- milk %>% select(species, kcal.per.g, perc.fat, perc.lactose) %>%
drop_na()
names(M) <- c("Species", "Milk_Energy", "Fat", "Lactose")
glimpse(M)
## Observations: 29 ## Variables: 4 ## $ Species <chr> "Eulemur fulvus", "E macaco", "E mongoz", "E rubrive… ## $ Milk_Energy <dbl> 0.49, 0.51, 0.46, 0.48, 0.60, 0.47, 0.56, 0.89, 0.91… ## $ Fat <dbl> 16.60, 19.27, 14.11, 14.91, 27.28, 21.22, 29.66, 53.… ## $ Lactose <dbl> 67.98, 63.82, 69.04, 71.91, 53.22, 55.20, 46.88, 30.…
Visualizing data
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Visualizing data
library(GGally) ggscatmat(as.data.frame(M), columns = 2:4)
Multiple regression
fm <- lm(Milk_Energy ~ Fat + Lactose, data = M) summary(fm)
## ## Call: ## lm(formula = Milk_Energy ~ Fat + Lactose, data = M) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.11350 -0.05047 0.01103 0.04649 0.12701 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 1.007372 0.211168 4.770 6.16e-05 *** ## Fat 0.001952 0.002533 0.771 0.44784 ## Lactose -0.008709 0.002575 -3.382 0.00229 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.06447 on 26 degrees of freedom ## Multiple R-squared: 0.8518, Adjusted R-squared: 0.8404 ## F-statistic: 74.74 on 2 and 26 DF, p-value: 1.657e-11
Visualizing multiple regression
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Estimate Fat coefficient
- Use
Lactoseto predictFat, which will take the effect ofLactoseout of the model when we predictMilk_Energy - Extract the residuals and add them onto the data.
fm_Lact <- lm(Fat ~ Lactose, data = M) M$resid_Lact <- residuals(fm_Lact) head(M)
## # A tibble: 6 x 5 ## Species Milk_Energy Fat Lactose resid_Lact ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 Eulemur fulvus 0.49 16.6 68.0 0.196 ## 2 E macaco 0.51 19.3 63.8 -1.12 ## 3 E mongoz 0.46 14.1 69.0 -1.28 ## 4 E rubriventer 0.48 14.9 71.9 2.27 ## 5 Lemur catta 0.6 27.3 53.2 -3.25 ## 6 Alouatta seniculus 0.47 21.2 55.2 -7.42
Estimate Fat coefficient
Estimate Fat coefficient
Estimate Fat coefficient
coef(lm(Milk_Energy ~ resid_Lact, data = M))
## (Intercept) resid_Lact ## 0.641724138 0.001952441
coef(fm)
## (Intercept) Fat Lactose ## 1.007371840 0.001952441 -0.008708827
Estimate Lactose coefficient
- Use
Fatto predictLactose, which will take the effect ofFatout of the model when we predictMilk_Energy - Extract the residuals and add them onto the data.frame
M.
fm_Fat <- lm(Lactose ~ Fat, data = M) M$resid_Fat <- residuals(fm_Fat)
Estimate Lactose coefficient
Estimate Lactose coefficient
Estimate Lactose coefficient
coef(lm(Milk_Energy ~ resid_Fat, data = M))
## (Intercept) resid_Fat ## 0.641724138 -0.008708827
coef(fm)
## (Intercept) Fat Lactose ## 1.007371840 0.001952441 -0.008708827
Compare
Compare
Multicollinearity
High correlation between predictors leaves little residual variation to be used for explaining the outcome variable.
Masking
Multiple predictors are useful for predicting outcomes when bivariate relationships with the response variable is not strong.
But:
- Associative relationships can be obscured when two predictors are somewhat correlated with one another.
Mammal milk data in a different context
Milk is a big energetic investment
- Is there a significant association between energy content of milk while controlling for neocortex size and body size?
- Do primates with larger brains produce significantly more nutritious milk so their offspring can grow quickly (because they must grow quickly) ?
Visualizing
Visualizing
Bivariate model of Neocortex
fm_Neo <- lm(Milk_Energy ~ Neocortex, data = M) summary(fm_Neo)
## ## Call: ## lm(formula = Milk_Energy ~ Neocortex, data = M) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.19027 -0.14693 -0.03744 0.15613 0.29959 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.353332 0.501120 0.705 0.492 ## Neocortex 0.004503 0.007389 0.609 0.551 ## ## Residual standard error: 0.1764 on 15 degrees of freedom ## Multiple R-squared: 0.02417, Adjusted R-squared: -0.04089 ## F-statistic: 0.3715 on 1 and 15 DF, p-value: 0.5513
Bivariate model of log Mass
fm_Mass <- lm(Milk_Energy ~ log_Mass, data = M) summary(fm_Mass)
## ## Call: ## lm(formula = Milk_Energy ~ log_Mass, data = M) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.26908 -0.09190 -0.03189 0.13180 0.30209 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.70516 0.05185 13.599 7.68e-10 *** ## log_Mass -0.03169 0.02160 -1.467 0.163 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.167 on 15 degrees of freedom ## Multiple R-squared: 0.1255, Adjusted R-squared: 0.0672 ## F-statistic: 2.153 on 1 and 15 DF, p-value: 0.163
Multivariate model
fm_Multi <- lm(Milk_Energy ~ Neocortex + log_Mass, data = M) summary(fm_Multi)
## ## Call: ## lm(formula = Milk_Energy ~ Neocortex + log_Mass, data = M) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.250574 -0.039212 0.000633 0.072997 0.201985 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -1.085254 0.515281 -2.106 0.05372 . ## Neocortex 0.027931 0.008015 3.485 0.00364 ** ## log_Mass -0.096402 0.024749 -3.895 0.00162 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1265 on 14 degrees of freedom ## Multiple R-squared: 0.5317, Adjusted R-squared: 0.4648 ## F-statistic: 7.948 on 2 and 14 DF, p-value: 0.004939
Interpretation
- Both coefficients go up
- Neocortex: \(0.005 \rightarrow 0.03\) (P = 0.004)
- log Mass: \(-0.03 \rightarrow -0.1\) (P = 0.002)
Regression asks (and answers):
- Do species that have high neocortex percentage for their mass have higher energy milk?
- Do species with high body mass for their neocortex percentage have higher energy milk?
Neocortex vs. log Mass
Milk Energy vs. Residual Mass
Quiz 07-3
Complete Quiz 07-3
Watch Lecture 07-4